Canonical Hamiltonian equations, real and complex, and a colloquium

Here are the canonical forms of Hamiltonian equations, in both the most familiar real form

\displaystyle \frac{d q_n}{d t} = \frac{\partial H}{\partial p_n}, \quad \frac{d p_n}{d t} = -\frac{\partial H}{\partial q_n}

and the complex form,

\displaystyle \frac{d z_n}{d t} = i \frac{\partial H}{\partial z_n^*}, \quad \frac{d z_n^*}{d t} = -i\frac{\partial H}{\partial z_n}

which is often conveniently in situations involving either Schroedinger’s equation or other conservative equations involving complex quantities, such as in optics.

The two forms are connected by the change of variables

\displaystyle z_n = \frac{q_n + i p_n}{\sqrt 2}, \; z_n^* = \frac{q_n - i p_n}{\sqrt 2}

where one has to allow the quantities q_n and p_n to have complex values.

Why this post?

  1. Because I can: posting to a blog hosted at WordPress is the only option I currently know of that allows mathematical notation to be used (via latex-like mark-up) when posting to a web-site from a mobile device like this iPad
  2. To do something work-related with this employer-provided iPad … though I do also use it for all my in-class PDF presentations
  3. To advertise my upcoming colloquium talk on numerical methods for solving such equations in a way that respects all conservation laws: Friday November 18; details at math.cofc.edu/colloquium
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4 Responses to Canonical Hamiltonian equations, real and complex, and a colloquium

  1. Anna says:

    why are the in line equations offset upward? Other than that, wordpress is a nice tool!

    • Aha: changing the layout theme to “Tarski” fixed the equations [Edit: not really! The bottom is still level with the surrounding text’s baseline; but somehow it seems better], and litend lightened the title font too. It seems that only some themes play well with the mathematical mark-up.

  2. Bryan -- TLT says:

    Looks good to me!

    — Bryan

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